Is 6/7 Bigger Than 5/9?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 7 and 9, and the least common denominator (LCD) is 63.

• Step 1. Convert \(\frac{6}{7}\): multiply numerator and denominator by 9.
  \[\frac{6}{7} = \frac{6 \times 9}{7 \times 9} = \frac{54}{63}\]
• Step 2. Convert \(\frac{5}{9}\): multiply numerator and denominator by 7.
  \[\frac{5}{9} = \frac{5 \times 7}{9 \times 7} = \frac{35}{63}\]
• Step 3. Compare the numerators: 54 vs 35.
• Step 4. Since 54 is greater than 35, we know that \(\frac{6}{7} > \frac{5}{9}\).

Method 2: Cross Multiplication

A quicker way to compare fractions is to cross-multiply. Multiply each numerator by the other fraction's denominator:

• Step 1. Multiply the numerator of the first fraction by the denominator of the second: \(6 \times 9 = 54\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(5 \times 7 = 35\).
• Step 3. Compare the products: \(54 > 35\), so \(\frac{6}{7} > \frac{5}{9}\).

Method 3: Decimal Comparison

Convert each fraction to a decimal by dividing the numerator by the denominator:

Since 0.857143 is greater than 0.555556, we confirm that \(\frac{6}{7} > \frac{5}{9}\). In percentage terms, \(\frac{6}{7}\) is 85.7143% and \(\frac{5}{9}\) is 55.5556%, a difference of 30.1587 percentage points.

Why Does This Work?

These fractions have different numerators and different denominators, so we can't compare them directly. By converting to a common denominator of 63, we're cutting both quantities into equal-sized pieces. Then 54 pieces vs 35 pieces is a straightforward comparison.